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31 IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
J. J. C. PITUBA | M. M. S. LACERDA
4. Numerical Applications
4.1 Plain Concrete Beam
In this first numerical application, previously performed by
Guello [15], we have considered the concrete beam without
any reinforcement bar described in Figure [1]. The beam whose
concrete has elasticity modulus Ec = 24700 MPa, is subjected to
two concentrated loads P applied at a distance of 0,225 m from
the symmetrical axes.
Table [1] contains the parameter values of both models employed
in this example. The parametric identification of the damage mod-
els has been done using a computational code developed by Pitu-
ba [11] based on error minimization procedure. The compression
and tension parameters values of the Pituba’s model have been
identified by numerical responses proposed by Guello [15] using
the Mazars’ model, as described in Figure [2].
Moreover, the parameters values associated to D3 have been
identified by numerical simulation of biaxial stress tests in concrete
specimens, Pituba [11]. It is important to note that the concrete
used for the identification of Y03, A 3 and B3 parameters is similar to
the one considered in this numerical application.
For the 1D analysis, bar elements with transversal section
stratified in layers are used, see Figure [3] where layer K can
represent concrete or reinforcement bar. A mesh with 40 ele-
ments and 20 layers has been considered. On the other hand,
a mesh with 120 constant strain quadrilateral (4 nodes) ele-
ments divided into 5 layers of 24 elements and placed in whole
extension of the concrete beam has been considered for the 2D
analysis, as described in Figure [4]. Note that in this numerical
example, the layer with finite elements in black representing the
reinforcement does not exist.
The numerical responses are displayed in Figure [5]. In the
2D analysis context, it can be observed that the difference be-
Table 1 – Parameter values – plain concrete beam
Mazars’ model Pituba’s model
Tension Compression Tension Compression
A = 0,7T A = 1,13C
-4 Y = 0,25x10 MPa01
-3 Y = 0,5x10 MPa02
-3 Y = 0,5x10 MPa03
B = 8000T B = 1250C A = 501 A = -0,92 A = -0,63
e = 0,000067d0
-1 B = 6700 MPa1
-1 B = 0,4MPa2
-1 B = 1305MPa3
Figure 3 – 1D finite element
Figure 4 – 2D finite element discretization
Z
Y
X
Z
Y
Z
X
Figure 5 – 1D and 2D numerical responses
for plain concrete beam